We study the nonlinear quantum master equation describing a laser under the mean field approximat... more We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.
Infinite Dimensional Analysis, Quantum Probability and Related Topics
We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purel... more We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.
The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free suba... more The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterization of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.
Summary We prove a Lévy type characterization theorem for the free Brownian motion and the free P... more Summary We prove a Lévy type characterization theorem for the free Brownian motion and the free Poisson process using martingale and convariance conditions and some assumption on fourth order conditional moments.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2016
We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS cond... more We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one.
Latin American Journal of Probability and Mathematical Statistics, Jul 12, 2012
We study stochastic evolution equations describing the dynamics of open quantum systems. First, u... more We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a sufficient condition for regularity of solutions to linear stochastic Schrödinger equations driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrödinger equations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2015
We show that the commutant of the range of the infinitesimal generator of a norm-continuous quant... more We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.
Quantum Optics has had an impressive development during the last two decades. Laser-based devices... more Quantum Optics has had an impressive development during the last two decades. Laser-based devices are invading our everyday life: printers, pointers, a new surgical technology replacing the old scalpel, transport of information, among many other applications which give the idea of a well established theory. Even though, various theoretical problems remain open. In particular Quantum Optics continues to inspire mathematicians in their own research. This survey is aimed at giving a taste of some mathematical problems raised by Quantum Optics within the field of Stochastic Analysis. Especially, we want to show the interplay between classical and non commutative stochastic differential equations associated to the so called master equations of Quantum Optics. Contents 1. Introduction 2 1.1. Notations 3 1.2. Deriving the master equation 4 2. The associated Quantum Stochastic Differential Equation 7 2.1. Quantum noises in continuous time 8 2.2. The master equation for the quantum cocycle 9 2.3. The Quantum Markov Flow 10 3. Unravelling: classical stochastic processes related to quantum flows 12 3.1. The algebra generated by the number operator 12 3.2. Position, momentum and their algebras 13 3.3. The evolution on A N , A q , A p. 15 4. Stationary state and the convergence towards the equilibrium 16 References 18 Research partially supported by FONDECYT grant 1960917 and the "Cátedra Presidencial en Análisis Estocástico y Física Matemática".
Comptes Rendus De L Academie Des Sciences Serie 1 Mathematique, 1995
Etant donne un semigroupe quantique (T t ; t ≥ 0) possedant un etat stationnaire normal fidele ρ,... more Etant donne un semigroupe quantique (T t ; t ≥ 0) possedant un etat stationnaire normal fidele ρ, cette Note etablit une condition suffisante pour que le semigroupe converge vers (tr ρ.) I lorsque t → ∞, au sens de la topologie faible w*, ou I designe l'operateur identite.
We give an explicit entropy production formula for a class of quantum Markov semigroups, arising ... more We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.
Indian Journal of Pure and Applied Mathematics, 2015
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochas... more A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schrödinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups.
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochas... more A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schroedinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov
It is a great pleasure to thank also R. Soto and the Editorial Board of "Proyecciones" for public... more It is a great pleasure to thank also R. Soto and the Editorial Board of "Proyecciones" for publication of this unusually lengthy article. 3 k=0 i −k v + i k u, (T x α)(v + i k u) it follows then that (1.2) holds for each v, u ∈ h. 4 implies 3. In fact 4 implies that (1.2) holds for each v, u in the dense subset of h linearly spanned by the total set. This linear span is obviously dense. 3 implies 2. Let (v n) n≥0 , (u n) n≥0 be two sequences of vectors in h such that the sequences (v n) n≥0 , (u n) n≥0 are square-summable. We must show that lim α n≥0 v n , (T x α)u n = n≥0 v n , (T x)u n. To this end, for every ε > 0, take an integer ν such that n>ν u n 2 < ε, n>ν v n 2 < ε. Remark. Notice that, under the assumptions of Theorem A.1, P (t)u = T (t)u for every u ∈ L 2 (IR d ; I C) ∩ C 0 0 (IR d ; IR) and t ≥ 0. In fact P (t) and T (t) are bounded operators coinciding on the dense (in L 2 (IR d ; I C) and C 0 0 (IR d ; IR)) subset C ∞ c (IR d ; IR). We refer to [43] for the proofs of the above results.
It is proved that every non-divergent classical pure birth process and every classical pure death... more It is proved that every non-divergent classical pure birth process and every classical pure death process can be realised as a quantum flow in Fock space.
The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free suba... more The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a naturaldecomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterisation of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.
We describe a rather general scheme for extending the flow of a classical Markov process to a qua... more We describe a rather general scheme for extending the flow of a classical Markov process to a quantum flow on the algebra of all bounded operator on an L 2 space of the state space. We discuss open problems and possible developments.
We study the nonlinear quantum master equation describing a laser under the mean field approximat... more We study the nonlinear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the nonlinear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schrödinger equation. To this end, we find a regular solution for the nonautonomous linear quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form, and we prove the uniqueness of the solution to the nonautonomous linear adjoint quantum master equation in Gorini–Kossakowski–Sudarshan–Lindblad form. Moreover, we obtain rigorously the Maxwell–Bloch equations from the mean field laser equation.
Infinite Dimensional Analysis, Quantum Probability and Related Topics
We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purel... more We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.
The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free suba... more The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterization of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.
Summary We prove a Lévy type characterization theorem for the free Brownian motion and the free P... more Summary We prove a Lévy type characterization theorem for the free Brownian motion and the free Poisson process using martingale and convariance conditions and some assumption on fourth order conditional moments.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2016
We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS cond... more We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one.
Latin American Journal of Probability and Mathematical Statistics, Jul 12, 2012
We study stochastic evolution equations describing the dynamics of open quantum systems. First, u... more We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a sufficient condition for regularity of solutions to linear stochastic Schrödinger equations driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrödinger equations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2015
We show that the commutant of the range of the infinitesimal generator of a norm-continuous quant... more We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.
Quantum Optics has had an impressive development during the last two decades. Laser-based devices... more Quantum Optics has had an impressive development during the last two decades. Laser-based devices are invading our everyday life: printers, pointers, a new surgical technology replacing the old scalpel, transport of information, among many other applications which give the idea of a well established theory. Even though, various theoretical problems remain open. In particular Quantum Optics continues to inspire mathematicians in their own research. This survey is aimed at giving a taste of some mathematical problems raised by Quantum Optics within the field of Stochastic Analysis. Especially, we want to show the interplay between classical and non commutative stochastic differential equations associated to the so called master equations of Quantum Optics. Contents 1. Introduction 2 1.1. Notations 3 1.2. Deriving the master equation 4 2. The associated Quantum Stochastic Differential Equation 7 2.1. Quantum noises in continuous time 8 2.2. The master equation for the quantum cocycle 9 2.3. The Quantum Markov Flow 10 3. Unravelling: classical stochastic processes related to quantum flows 12 3.1. The algebra generated by the number operator 12 3.2. Position, momentum and their algebras 13 3.3. The evolution on A N , A q , A p. 15 4. Stationary state and the convergence towards the equilibrium 16 References 18 Research partially supported by FONDECYT grant 1960917 and the "Cátedra Presidencial en Análisis Estocástico y Física Matemática".
Comptes Rendus De L Academie Des Sciences Serie 1 Mathematique, 1995
Etant donne un semigroupe quantique (T t ; t ≥ 0) possedant un etat stationnaire normal fidele ρ,... more Etant donne un semigroupe quantique (T t ; t ≥ 0) possedant un etat stationnaire normal fidele ρ, cette Note etablit une condition suffisante pour que le semigroupe converge vers (tr ρ.) I lorsque t → ∞, au sens de la topologie faible w*, ou I designe l'operateur identite.
We give an explicit entropy production formula for a class of quantum Markov semigroups, arising ... more We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.
Indian Journal of Pure and Applied Mathematics, 2015
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochas... more A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schrödinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov semigroup, nevertheless, they are equivalent for some important classes of semigroups.
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochas... more A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schroedinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov
It is a great pleasure to thank also R. Soto and the Editorial Board of "Proyecciones" for public... more It is a great pleasure to thank also R. Soto and the Editorial Board of "Proyecciones" for publication of this unusually lengthy article. 3 k=0 i −k v + i k u, (T x α)(v + i k u) it follows then that (1.2) holds for each v, u ∈ h. 4 implies 3. In fact 4 implies that (1.2) holds for each v, u in the dense subset of h linearly spanned by the total set. This linear span is obviously dense. 3 implies 2. Let (v n) n≥0 , (u n) n≥0 be two sequences of vectors in h such that the sequences (v n) n≥0 , (u n) n≥0 are square-summable. We must show that lim α n≥0 v n , (T x α)u n = n≥0 v n , (T x)u n. To this end, for every ε > 0, take an integer ν such that n>ν u n 2 < ε, n>ν v n 2 < ε. Remark. Notice that, under the assumptions of Theorem A.1, P (t)u = T (t)u for every u ∈ L 2 (IR d ; I C) ∩ C 0 0 (IR d ; IR) and t ≥ 0. In fact P (t) and T (t) are bounded operators coinciding on the dense (in L 2 (IR d ; I C) and C 0 0 (IR d ; IR)) subset C ∞ c (IR d ; IR). We refer to [43] for the proofs of the above results.
It is proved that every non-divergent classical pure birth process and every classical pure death... more It is proved that every non-divergent classical pure birth process and every classical pure death process can be realised as a quantum flow in Fock space.
The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free suba... more The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a naturaldecomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. This leads to a new characterisation of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.
We describe a rather general scheme for extending the flow of a classical Markov process to a qua... more We describe a rather general scheme for extending the flow of a classical Markov process to a quantum flow on the algebra of all bounded operator on an L 2 space of the state space. We discuss open problems and possible developments.
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